# What is symbol in signal processing?

I am new to signal processing and have been trying to understand the terminology. I came across this statement: "Assuming the signal modulation $$\mathfrak{M}$$ has an alphabet A of M symbols and the symbol $$A_{m}$$ having the equal probability to be transmitted". What is $$A_{m}$$ exactly here or what kind of array will it be? Can anyone help me understand this with any modulation example?

A symbol is a symbolic representation of a baseband signal in digital communication. Imagine you have 2 bits with equal probability 0 and 1. For 0 you transmit $$s_0(t)=+A$$ for $$0 \le t \le T$$. For 1 you transmit $$s_1(t)=-A$$ for $$0 \le t \le T$$. You are basically varying the amplitudes of a basic vector signal which is rectangle in this case $$r(t) = A$$ for $$0 \le t \le T$$. Here $$s_0(t) = 1 \times r(t)$$ and $$s_1(t)=-1 \times r(t)$$. The 'symbols' -1 and +1 are the scalars which multiply the fundamental signal. $$A_m = \pm 1$$.
If you vary phase instead of amplitude, you get schemes like BPSK(technically still AM), QPSK,8-PSK,...,M-PSK in general. For QPSK your inphase and quadrature bits both decide the phase offset. $$x_{qpsk}(t) = (-1)^b_I(1/\sqrt{2})\cos(\omega_c t)-(-1)^b_Q(1/\sqrt{2})\sin(\omega_c t)$$ which results in baseband equivalent phase offset of $$i\times \pi/4$$ where $$i \in \{0,1,2,3\}$$. $$M=4$$.$$A_m = \pm 1 \pm j$$.
For QAM, you vary both amplitude and phase, resulting in symbols which represent both phase as well as amplitude change. Example 16-QAM where $$M=16$$. 